If `(sec theta + tan theta)/(sec theta - tan theta) = 5/3`, then `sin theta` is equal to

Options

A. `2//3`
B. `3//4`
C. `1//4`
D. `1//3`

Answer

To solve the equation \(\frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{5}{3}\), we will follow these steps:

### Step 1: Rewrite sec and tan in terms of sin and cos
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ \frac{\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}} = \frac{5}{3} \]

### Step 2: Simplify the equation
The equation simplifies to:
\[ \frac{\frac{1 + \sin \theta}{\cos \theta}}{\frac{1 - \sin \theta}{\cos \theta}} = \frac{5}{3} \]
This can be further simplified by canceling \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \frac{1 + \sin \theta}{1 - \sin \theta} = \frac{5}{3} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 3(1 + \sin \theta) = 5(1 - \sin \theta) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 3 + 3\sin \theta = 5 - 5\sin \theta \]

### Step 5: Combine like terms
Rearranging the equation to combine all \(\sin \theta\) terms on one side gives:
\[ 3\sin \theta + 5\sin \theta = 5 - 3 \]
This simplifies to:
\[ 8\sin \theta = 2 \]

### Step 6: Solve for \(\sin \theta\)
Dividing both sides by 8 gives:
\[ \sin \theta = \frac{2}{8} = \frac{1}{4} \]

### Final Answer
Thus, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{1}{4} \]

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